proving the existence of a minimizer of this functional - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T13:57:36Z http://mathoverflow.net/feeds/question/109560 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109560/proving-the-existence-of-a-minimizer-of-this-functional proving the existence of a minimizer of this functional Cyan 2012-10-13T21:34:08Z 2012-10-17T22:07:45Z <p>I seek the minimum of a certain functional which is always strictly greater than zero. The Euler-Lagrange equation is a "zeroth-order differential equation", that is, an implicit equation that a stationary function must satisfy. Assuming that there is only one such function, I seek to prove that it is in fact the minimizer of the functional. </p> <p>Since the first variation doesn't depend on the derivative of the functional's argument, variations of all orders are zero at the stationary function, so I can't use the second variation test. I've looked a bit into direct methods, but I'm not a mathematician and I quickly got lost in the discussion of topologies of function spaces. The literature is so large and I am so unfamiliar with it that I'm having trouble sorting out exactly what assumptions are sufficient for the existence of a minimizer.</p> <p>Here's my functional, $\mathcal{A}[f]$: </p> <p>Let $w:\mathbb{R}\rightarrow\mathbb{R}^+$ be a weight function, let $g(x, y) \in C^2(\mathbb{R}^2)$, and let $u_{f} (x) = g(x, f(x))$. Then</p> <p>$$\mathcal{A}[f] = \int^{x_1}_{x_0} f(x)[w(u_f(x))\frac{\mathrm{d}u}{\mathrm{d}x}-w(x)]\mathrm{d}x.$$ </p> <p>The Euler-Lagrange equation is </p> <p>$$\frac{w(x)}{w(u_{f}(x))} = \left. \frac{\partial g}{\partial x} \right|_{y=f(x)}.$$</p> http://mathoverflow.net/questions/109560/proving-the-existence-of-a-minimizer-of-this-functional/109958#109958 Answer by dcs24 for proving the existence of a minimizer of this functional dcs24 2012-10-17T22:07:45Z 2012-10-17T22:07:45Z <p>I haven't verified your calculation of the E-L equation. But if the Jacobi-field equation (second variation) is degenerate then you have to show that $A[f+g] \leq A[f]$ for either all $g$ or, endowing your space of functions with a suitable norm for $g \leq \varepsilon$ for some $\varepsilon > 0$.</p> <p>My feeling is to, assume your weight function is sufficiently smooth ($C^2$ or more), use Taylors theorem to expand out your nonlinear $w$ and $g$, bound out the highest order terms (the remainders) and look at the remaining problem.</p>